40 
MR. LUBBOCK’S RESEARCHES 
3 . j 3 a- . i 15 a 3 
~2 ~ ~2 ^ Sln2 "2 cos ( 2 t-2y)- -g- — t cos (* + 2y) 
[76] ' [77] 
[78] 
15 a 3 . i 
— -g cos (3 t — 2y) 
[79] 
Let ~ -= 1 + ecos 0 + A:) t + e — + e 2 (l + r 48 ) cos ^2 n (1 + k 2 ) i + 2 s — 2 
+ e, 9 ?-^ cos ^2 rc (1 + * 49 ) t + 2e — 2ro^ + r 0 + r, cos (n £ + n ( i + s — e ( ) 
+ r 2 cos (2 « £ — 2 n t t + 2 £ — 2 £,) + &c. + er 6 cos (re/ + £, — m) + &c. 
^£l_£+ii + 2/Jj! + r(' d *')=0 
2d* 2 r a J \dr / 
Let & . -i denote that part of ^ which depends on the first power of the dis- 
turbing force. It is more simple to obtain S . ~ from the above differential equa- 
tion than and the circumstance that the elliptic value of r 3 does not contain 
any term e 2 cos (2 n t + 2 s — 2 7s), gives an additional facility, 
l 
-„ S J + 2fAR + r(™)=0 
d 2 .r 3 S'.- 
d l 2 
When the disturbing force is neglected 
= a- ^ 1 + 3 e- ^ 1 + L^_ 3 e ^i ^ cos (re t + e — m) — e 4 cos (2 re t + 2 £ — 2 m) 
+ — cos (3 re < + 3 £ — 3 rer) + — cos (4 n t + 4 £ 
o 8 
— 4 m) | 
Integrating the above differential equation by the method of indeterminate 
co-efficients, q being the co-efficient of the rath argument in the development of 
— r o + — a 7 0 = 0 
l J - 
{ ( 1 + 3 e-) U - *±L (r 6 + r 8 ) j - r, + ^ a 9l = 0 
(2 ” { (l + 3 e ') r « ~ ^ ( r u + Us) } ~ U + y a <ii = 
0 
